Several schemes for discretization of first and second derivatives are available in Smoothed Particle Hydrodynamics (SPH). Here, four schemes for approximation of the first derivative and three ...schemes for the second derivative are examined using a theoretical analysis based on Taylor series expansion both for regular and irregular particle distributions. Estimation of terms in the truncation errors shows that only the renormalized (the first-order consistent) scheme has acceptable convergence properties to approximate the first derivative. None of the second derivative schemes has the first-order consistency. Therefore, they converge only when the particle spacing decreases much faster than the smoothing length of the kernel function.
In addition, using a modified renormalization tensor, a new SPH scheme is presented for approximating second derivatives that has the property of first-order consistency. To assess the computational performance of the proposed scheme, it is compared with the best available schemes when applied to a 2D heat equation. The numerical results show at least one order of magnitude improvement in accuracy when the new scheme is used. In addition, the new scheme has higher-order convergence rate on regular particle arrangements even for the case of only four particles in the neighborhood of each particle.
A weakly compressible smoothed particle hydrodynamics (WCSPH) method is used along with a new no-slip boundary condition to simulate movement of rigid bodies in incompressible Newtonian fluid flows. ...It is shown that the new boundary treatment method helps to efficiently calculate the hydrodynamic interaction forces acting on moving bodies. To compensate the effect of truncated compact support near solid boundaries, the method needs specific consistent renormalized schemes for the first and second-order spatial derivatives. In order to resolve the problem of spurious pressure oscillations in the WCSPH method, a modification to the continuity equation is used which improves the stability of the numerical method. The performance of the proposed method is assessed by solving a number of two-dimensional low-Reynolds fluid flow problems containing circular solid bodies. Wherever possible, the results are compared with the available numerical data.
► A weakly compressible SPH method was proposed for solving particulate flow problems. ► A new solid boundary treatment approach is applied to the moving boundaries. ► Spurious pressure oscillations are reduced by using a modified continuity equation. ► Consistent schemes of the first and second order spatial derivatives are used.
•Two density-based iterative SPH methods are proposed for incompressible flows.•They eliminate non-physical pressure waves in density-based schemes.•Both proposed DTS-SPH and AL-SPH methods generate ...smooth pressure fields.
In this study, we have introduced two new iterative density-based Smoothed Particle Hydrodynamics (SPH) methods to model incompressible flows, namely, preconditioned dual time-stepping, and augmented Lagrangian method. The performance of these new methods are compared with each other and also with a modified version of the well-known weakly compressible SPH (WC-SPH) method through solving three carefully chosen incompressible flow problems: a laminar incompressible channel flow over a backward-facing step, a 2D stiff pressure decay problem and Taylor-Green vortices flow. For the first test problem, the results are compared with available data in literature. Moreover, it is observed that the two iterative methods provide a better accuracy in terms of smoother pressure field and also smaller magnitude of the velocity divergence across the computational domain. In the second test problem, it is shown that the preconditioned dual time-stepping and the augmented Lagrangian SPH methods yield rather smooth pressure fields, and converge to the exact solution, while the pressure field computed by the WC-SPH method oscillates even after very long time. As for the third test case, the iterative methods are compared with WC-SPH method for different iteration numbers and particle resolutions.
Improving the accuracy and convergence rate of the smoothed particle hydrodynamics (SPH) method is still one of the challenges that researchers are trying to achieve. In this research, the first- and ...second-order convergence schemes have been used for the first and second spatial derivatives, respectively, in continuity and linear momentum conservation equations to increase accuracy and convergence rate. These schemes have been implemented in the open-source code of DualSPHysics, one of the prominent and powerful software of the SPH to calculate the particle velocity and pressure of the Taylor–Green vorticity benchmark problem. Both the standard and the improved solvers were evaluated with the same numerical settings, different particle spacing, and Reynolds numbers of 100 and 1000 with the same dimensionless time. Up to three orders of magnitude improvement in the accuracy were achieved in the results of the improved solver, as well as a significant increase in convergence rate.
In this paper, a pressure splitting formulation is proposed for Weakly Compressible SPH (WC-SPH) method and its capability in the suppression of the spurious oscillations is studied by conducting a ...stability analysis. The proposed formulation is implemented within the framework of a consistent SPH method. The predictions from the theoretical analysis are verified by the results of numerical test-cases. This method is applied to the incompressible fluid flow around periodic array of circular cylinders. The accuracy and the convergence of the results are investigated for benchmark problems. The results are also compared with those of the conventional WC-SPH method. In a similar test-case, the effects of the artificial speed of sound on the evolution of the transient solution and the occurrence of the spurious oscillations in the steady state are studied, and compared for both the conventional and the proposed WC-SPH formulations. The improvements in the evaluation of the pressure field, due to the proposed pressure splitting formulation, are shown for both a vanishing Reynolds number and a finite Reynolds number of Re∼O(1).
A direct numerical simulation approach is utilized to understand the oscillatory shear rheology of a confined suspension of magnetic chains formed by paramagnetic circular cylinders under the ...influence of an external magnetic field. The common assumption of gap-spanning chains made in the literature is relaxed in this work, so that a fully suspended (periodic) array of magnetic chains is formed. In this sense, the effective rheological parameters are only influenced through a layer of fluid adjacent to the walls. All tests are conducted at very low but finite particle Reynolds numbers, and typical inertial effects are discussed. The main aim of the present study is to investigate the apparent viscoelasticity of the system as a function of the external magnetic field and frequency of the input strain. This work concentrates on cases with large blockage ratio in order to have pronounced viscoelastic behaviours.
► The movement of the solid bodies is studied in an Oldroyd-B shear flow. ► An improved weakly compressible SPH method is used. ► The effect of fluid elasticity on the migration mechanism of the ...solid bodies is investigated. ► The hydrodynamic interaction between two solid bodies is studied.
An explicit weakly compressible SPH method is introduced to study movement of suspended solid bodies in Oldroyd-B fluid flows. The proposed formulation does not need further stabilizing treatments and can be efficiently employed to study particulate flows with Deborah to Reynolds number ratios up to around 10. A modified boundary treatment technique is also presented which helps to deal with the movement of solid particles in the flow. The technique is computationally efficient and gives an improved evaluation of fluid-solid interaction forces.
A number of test cases are solved to show performance of the proposed method in simulating particulate viscoelastic flows containing circular and non-circular cylinders. The effect of Deborah number on the particle trajectory has been investigated.
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